MA3343 Groups. Week 5: Centres and Centralizers

Welcome to our activities for Week 5 (the week of October 3). We have two new concepts this week, closely related: the centre of a group, and the centralizer of an element in a group.

Both are related to the concept of commutativity. A quick summary here; the full details are in the first half of Section 2.2 of the lecture notes.

Recall that a binary operation * is commutative if a*b = b*a for all relevant elements a and b, and that a group is called abelian if its operation is commutative. As we know, groups can be abelian (like the group of integers under addition) or non-abelian (like the groups of symmetries of regular polygons). A group is non-abelian if it has even one pair of elements that don’t commute with each other under the group operation. A non-abelian group might have lots of elements that commute with each other, or lots of elements that commute with all other elements, or it might have few. The concepts of the centre and the centralizer of an element capture to some extent how far away a group is from being abelian, or to what extent an element fails to commute with others.

• The centre of a group is the subset of the group consisting of those elements that commute with all elements in the group.
• The centralizer of an element of a group is the subset consisting of all elements which commute with that one.

This week’s lectures discuss these concepts, highlight some properties of the centre and centralizers (the principal one is that both are subgroups). Slides for this week’s lectures are here.

Here are (old) video versions of this week’s lectures, for use as backup and in case anyone can’t make it in person. As usual, please ignore any references to details (like assessment arrangements) that are not current this year.